Test 2
Suppose that E and F are two events and that P(E and F) = 0.1 and P(E) = 0.2. What is P(F|E)?
0.5 because 0.1 / 0.2 = 0.5 If E and F are any two events, then the probability of event F occurring, given the occurrence of event E, is found by dividing the probability of E and F by the probability of E. P(F|E) = P(E and F) / P(E)
Find the value of the permutation of 19P0
19P0 = 1 use formula: nPr = n! / (n - r)! 19! / (19 - 0)! --> 19! / 19! = (use calculator)
Find the value of the permutation of 4P2
4P2 = 12 use formula: nPr = n! / (n - r)! 4! / (4 - 2)! --> 4! / 2! = 12 (use calculator)
What does it mean for an event to be unusual? Why should the cutoff for identifying unusual events not always be 0.05?
An event is unusual if it has a low probability of occurring. The choice of a cutoff should consider the context of the problem.
The probability that a randomly selected individual in a country earns more than $75,000 per year is 7.5%. The probability that a randomly selected individual in the country earns more than $75,000 per year, given that the individual has earned a bachelor's degree, is 21.5%. Are the events "earn more than $75,000 per year" and "earned a bachelor's degree" independent?
No
Data Values: .25, .25, .15, .15, - .3, .2 Determine why it is not a probability model?
This is not a probability model because at least one probability is less than 0. note: one probability is - .3
True or False: In a probability model, the sum of the probabilities of all outcomes must equal 1.
True
x-values: y-values 0: 0.31 1: 0.28 2: 0.16 3: 0.11 4: 0.14 Is the distribution a discrete probability distribution?
Yes, because the sum of the probabilities is equal to 1 and each probability is between 0 and 1, inclusive.
Determine whether the following probability experiment represents a binomial experiment and explain the reason for your answer. Three cards are selected from a standard 52-card deck without replacement. The number of clubs selected is recorded.
No, because the trials of the experiment are not independent and the probability of success differs from trial to trial.
Determine if the following probability experiment represents a binomial experiment. A random sample of 80 professional athletes is obtained, and the individuals selected are asked to state their ages.
No, this probability experiment does not represent a binomial experiment because the variable is continuous, and there are not two mutually exclusive outcomes.
Determine whether the following probability experiment represents a binomial experiment and explain the reason for your answer. An experimental drug is administered to 160 randomly selected individuals, with the number of individuals responding favorably recorded.
Yes, because the experiment satisfies all the criteria for a binomial experiment.
About 12% of the population of a large country is allergic to pollen. If two people are randomly selected, what is the probability both are allergic to pollen? What is the probability at least one is allergic to pollen? Assume the events are independent. (a) The probability that both will be allergic to pollen is ??? (b) The probability that at least one person is allergic to pollen is ???
(a) .0144 because .12 x .12 = .0144 note: multiply the probability of one person, twice (b) .2256 P(at least 1 allergic) = 1 − P(neither allergic) 1 - (.88 x .88) --> 1 - .7744 = .2256
(a) probability that the card drawn from a standard 52-card deck is a queen is ??? (b) The probability that the card drawn from a standard 52-card deck is a queen, given that this card is court, is ???
(a) .077 because 4 / 52 = .077 note: there are 4 queens in a deck (b) .333 because 4 / 12 = .333 note: there are 12 quarts in a deck (4 of each: kings, queens, and jacks)
A test to determine whether a certain antibody is present is 99.7% effective. This means that the test will accurately come back negative if the antibody is not present (in the test subject) 99.7% of the time. The probability of a test coming back positive when the antibody is not present (a false positive) is 0.003. Suppose the test is given to four randomly selected people who do not have the antibody. (a) What is the probability that the test comes back negative for all four people? (b) What is the probability that the test comes back positive for at least one of the four people?
(a) .9881 because .997^4 = .9881 (b) .0119 because 1 - .9881 = .0119 P(at least one positive) = 1 - (all tests are negative)
(a) What is the probability of an event that is impossible? (b) Suppose that a probability is approximated to be zero based on empirical results. Does this mean that the event is impossible?
(a) 0 (zero) (b) no note: The probability of an event E is approximately the number of times event E is observed divided by the number of repetitions of the experiment, as shown below. Just because the event is not observed, does not mean that the event is impossible.
Determine whether the events E and F are independent or dependent. Justify your answer. (a) E: A person living at least 70 years. F: The same person regularly handling venomous snakes. (b) E: A randomly selected person coloring her hair black. F: A different randomly selected person coloring her hair blond. (c) E: The rapid spread of a cocoa plant disease. F: The price of chocolate.
(a) E and F are dependent because regularly handling venomous snakes can affect the probability of a person living at least 70 years. (b) E cannot affect F and vice versa because the people were randomly selected, so the events are independent. (c) The rapid spread of a cocoa plant disease could affect the price of chocolate, so E and F are dependent.
A probability experiment is conducted in which the sample space of the experiment is S={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, event E={2, 3, 4, 5} and event G={6, 7, 8, 9}. Assume that each outcome is equally likely. (a) List the outcomes in E and G. (b) Are E and G mutually exclusive?
(a) E and G = {} note: they share no variables in common (b) Yes, because the events E and G have no outcomes in common.
Match the linear correlation coefficient to the scatter diagram. The scales on the x- and y-axis are the same for each scatter diagram. (a) r = −0.969 (b) r = −1 (c) r = - 0.049
(a) Graph 2: dots go negative, not exact line (b) Graph 1: dots go in a negative line exactly (c) Graph 3: barley noticable trend/line
For the accompanying data set, (a) draw a scatter diagram of the data, (b) by hand, compute the correlation coefficient, and (c) determine whether there is a linear relation between x and y. (a) Draw a scatter diagram of the data. (b) By hand, compute the correlation coefficient. The correlation coefficient is ??? (c) Because the correlation coefficient is ??? and the absolute value of the correlation coefficient, ??? is ??? than the critical value for this data set, ???, ??? linear relation exists between x and y.
(a) Look at points and plot them, choose best graph (b) - .496 (use calculator) (c) negative, .496, not greater, .878, no - note: it asks for the absolute value, never a negative - note: if correlation coefficient < critical value then there is no linear relation
For the accompanying data set, (a) draw a scatter diagram of the data, (b) by hand, compute the correlation coefficient, and (c) determine whether there is a linear relation between x and y. (a) Draw a scatter diagram of the data. (b) By hand, compute the correlation coefficient. The correlation coefficient is ??? (c) Because the correlation coefficient is ??? and the absolute value of the correlation coefficient, ??? is ??? than the critical value for this data set, ???, ??? linear relation exists between x and y.
(a) Look at points and plot them, choose best graph (b) .943 (use calculator) (c) positive, .943, greater, .878, a positive - find critical value by looking at given table and looking at the y-value for the number of variables
Data Values: .2, .1, .25, .25, .15 (a) Is the table above an example of a probability model? (b) What do we call the outcome "blue"?
(a) No, because the probabilities do not sum to 1. note: add up probabilities; this equaled .95 not 1 (b) Impossible note: blue's probability is 0
(a) If the pediatrician wants to use height to predict head circumference, determine which variable is the explanatory variable and which is the response variable. (b) Draw a scatter diagram. Which represents the data? (c) Compute the linear correlation coefficient between the height and head circumference of a child. (d) Does a linear relation exist between height and head circumference? (e) The new linear correlation coefficient is r = ??? The conversion to centimeters had ???
(a) The explanatory variable is height and the response variable is head circumference. (b) Graph Description: Basic positive trend, match points to table (c) r = .893 (use calculator) (d) Yes, the variables height and head circumference are positively associated because r is positive and the absolute value of the correlation coefficient is greater than the critical value, .707 (e) .893, has no effect on r note: conversion of values will not affect correlation coefficient
Researchers initiated a long-term study of the population of American black bears. One aspect of the study was to develop a model that could be used to predict a bear's weight (since it is not practical to weigh bears in the field). One variable thought to be related to weight is the length of the bear. The accompanying data represent the lengths and weights of 12 American black bears. (a) Which variable is the explanatory variable based on the goals of the research? (b) Draw a scatter diagram of the data. (c) The linear correlation coefficient between weight & length is ??? (d) The variables weight of the bear and length of the bear are ???associated because r is ??? and the absolute value of the correlation coefficient, ??? is ??? than the critical value, ???
(a) The length of the bear note: the length will determine the weight (response) (b) Look at basic trend, and discover the highest and lowest points to compare to choices (c) .718 (use calculator) (d) positively, positive, .718, greater, .576
(a) Is the number of free-throw attempts before the first shot is made discrete or continuous? (b) Is the distance a baseball travels in the air after being hit discrete or continuous?
(a) The random variable is discrete. The possible values are x=0, 1, 2,... (b) The random variable is continuous. The possible values are d>0.
(a) Is the number of people with blood type A in a random sample of 31 people discrete or continuous? (b) Is the time it takes for a light bulb to burn out discrete or continuous?
(a) The random variable is discrete. The possible values are x=0, 1, 2,...31 (b) The random variable is continuous. The possible values are t>0.
(a) List all the combinations of five objects x, y, z, s, and t taken two at a time. (b) What is 5C2?
(a) xy, xz, xs, xt, yz, ys, yt, zs, zt, st note: A combination is a collection, without regard to order, of n distinct objects without repetition. (b) 10 use formula: nCr = n! / r!(n - r)! 5! / 2! (5 - 3)! --> 5! / 2!(2!) = 10
(a) List all the permutations of four objects x, y, z, and s taken two at a time without repetition. (b) What is 4P2?
(a) xy, xz, xs, yx, yz, ys, zx, zy, zs, sx, sy, sz (b) 12
An engineer wants to determine how the weight of a car, x, affects gas mileage, y. The following data represent the weights of various cars and their miles per gallon. Weight (x) : Miles Per Gallon (y) 2510: 23.2 2940: 19.2 3295: 19.9 3875: 16.6 4225: 10.5 (a) Find the least-squares regression line treating weight as the explanatory variable and miles per gallon as the response variable. (b) Interpret the slope and intercept, if appropriate. (c) Predict the miles per gallon of car C (3295) and compute the residual. Is the miles per gallon of this car above average or below average for cars of this weight? The predicted value is ??? miles per gallon The residual is ??? miles per gallon. Is the value above or below average? (d) Which of the following represents the data with the residual shown?
(a) y = - .00639x + 39.4 (use calculator) (b) The slope indicates the mean change in miles per gallon for an increase of 1 pound in weight. It is not appropriate to interpret the y-intercept because it does not make sense to talk about a car that weighs 0 pounds. (c) 18.34, 1.56, it is above average note: plug 3295 into the discovered least-squares regression line note: 19.9 - 18.34 = 1.56 (observed - predicted) note: "it" refers to the value in the table (d) Input table points into L1 and L2, input least-squares regression line into y= and graph both to compare to choices
(a) Find the least-squares regression line treating the number of absences, x, as the explanatory variable and the final grade, y, as the response variable. (b) Interpret the slope and y-intercept, if appropriate. (c) Predict the final grade for a student who misses five class periods and compute the residual. Is the observed final grade above or below average for this number of absences? The predicted final grade is ??? This observation has a residual of ??? which indicates that the final grade is ??? average. (d) Draw the least-squares regression line on the scatter diagram of the data. (e) Would it be reasonable to use the least-squares regression line to predict the final grade for a student who has missed 15 class periods?
(a) y = - 3.085x +89.335 (b) Slope: For every day absent, the final grade falls by 3.085 on average. Y-Int: For zero days absent, the final score is predicted to be 89.335. (c) 73.9, - .1, below note: plug 5 in found equation and then do observed value minus predicted value (what we just found) (d) Graph both using calculator (e) No, 15 missed classes is outside of the scope model
A golf ball is selected at random from a golf bag. If the golf bag contains 6 type A balls, 8 type B balls, and 7 type C balls, find the probability that the golf ball is not a type A ball. The probability that the golf ball is not a type A ball is ???
.714 note: add up everything but the A values and divide that by the total occurrences (8 + 7) / (6 + 7 +8) = .714
A probability experiment is conducted in which the sample space of the experiment is S={5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, event F = {6, 7, 8, 9, 10, 11}, and event G = {10, 11, 12, 13}. Assume that each outcome is equally likely. (a) List the outcomes in F or G. (b) Find P(F or G) by counting the number of outcomes in F or G. (c) Determine P(F or G) using the general addition rule.
(a) {6, 7, 8, 9, 10, 11, 12, 13} note: "or" implies that you list the variables of all sets without repeating (b) .667 note: find the number of values in the set F or G and divide that by the total set (S) which is 8/12 = .667 (c) P(F or G) = .5 + .333 − .167 = .666 use formula: P(F or G)= P(F) + P(G) − P(F and G) P(F) = variables in F / total variable count (S) P(F and G) = variables in common / total variable
A probability experiment is conducted in which the sample space of the experiment is S= {3,4,5,6,7,8,9,10,11,12,13,14}. Event E= {4,5,6,7,8,9} and event F= {8,9,10,11}. List the outcomes in E and F. Are E and F mutually exclusive? (a) List the outcomes in E and F (b) Are E and F mutually exclusive?
(a) {8,9} note: list values which they have in common (b) No. E and F have outcomes in common note: Two events are disjoint if they have no outcomes in common. Another name for disjoint events is mutually exclusive events.
Find probability P(E or F) if E and F are mutually exclusive, P(E) = 0.25, and P(F) = 0.47. The probability P(E or F) is ???
.72 note: Use the general addition rule to find the probability. The rule states that for any two events E and F, P(E or F) = P(E) + P(F) − P(E and F) .25 + .47 - 0 = .72 - we know P(E and F) are 0 because they are mutually exclusive and therefore have nothing in common
Suppose that E and F are two events and that N(E and F)=410 and N(E)=510. What is P(F|E)?
.804 because 501 / 401 = .804
For the month of June in a certain city, 95% of the days are cloudy. Also in the month of June in the same city, 23% of the days are cloudy and foggy. What is the probability that a randomly selected day in June will be foggy if it is cloudy?
.242 because .23 / .95 = .242 note: P(F|E) = P(E and F) / P(E) so the probability that it is cloudy and foggy divided by probability that it is cloudy
Find probability of the indicated event if P(E) = 0.30 and P(F) = 0.35. Find P(E and F) if P(E or F) = 0.40 P(E and F) = ???
.25 note: Use the general addition rule to find the probability. The rule states that for any two events E and F, P(E or F) = P(E) + P(F) − P(E and F) .40 = .30 + .35 - X --> .40 = .65 - X --> X = .25
Suppose events E and F are independent, P(E) = 0.6, and P(F) = 0.7. What is the P(E and F)? The probability P(E and F) is ???
.42 note: multiply the values due to them being independent so .6 x .7 = .42
A golf ball is selected at random from a golf bag. If the golf bag contains 7 orange balls, 4 green balls, and 13 yellow balls, find the probability of the following event. The golf ball is orange or green. The probability that the golf ball is orange or green is ???
.458 note: add up the values of orange and green and divide it by the total occurrences 7 + 4 / 24 = .458
Find the probability P(E^c) if P(E)=0.38. The probability P(E^c) is ???
.62 note: if E represents any event and E^c represents the complement of E, then the probability PE^c is given by the formula below. P(E^c) = 1 − P(E) 1 - .38 = .62
Find probability of the indicated event if P(E) = 0.35 and P(F) = 0.35. Find P(E or F) if P(E and F) = 0.05. P(E or F) = ???
.65 note: Use the general addition rule to find the probability. The rule states that for any two events E and F, P(E or F) = P(E) + P(F) − P(E and F) .35 + .35 - .05 = .65
Find the value of the permutation of 4P4
4P4 = 24 use formula: nPr = n! / (n - r)! 4! / (4 - 4)! --> 4! / 0! = 24 (use calculator)
Notation P(F|E) means the probability of event ??? given event ???
F, E
Determine if the following statement is true or false. When two events are disjoint, they are also independent.
False note: Two events are disjoint if they have no outcomes in common. Independence means that one event occurring does not affect the probability of the other event occurring. Therefore, knowing two events are disjoint means that the events are not independent.
Match the linear correlation coefficient to the scatter diagram. r= - 0.93
Graph Description: Dots have a trend of going down as you move right (negative) and follows a line shape very well
x-values: y-values 10: 0.36 20: 0.34 30: 0.31 40: 0.15 50: -0.16 Is the distribution a discrete probability distribution?
No, because each probability is not between 0 and 1, inclusive.
x-values: y-values 0: .5 1: .5 2: .5 3: .5 4: .5 Is the distribution a discrete probability distribution?
No, because the sum of the probabilities is not equal to 1.
A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n = 15, p = 0.7, x = 12 What is P(12)?
P(12) = .17 use formula: P(x) = nCx p^x (1 − p)^n−x P(12) = 15C12 (.7)^12 (1 - .7)^15 - 12 --> P(12) = [ 15! / 12! (15 - 12)! ] (.7)^12 (.3)^3 = .17
A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n = 20, p = 0.01, x = 2 What is P(2)?
P(2) = .0159 use formula: P(x) = nCx p^x (1 − p)^n−x P(2) = 20C2 (0.01)^2 (1 - 0.01)^20 - 2 P(2) = [ 20! / 2! (20 - 2)! ] (0.01)^2 (.99)^18 = .0159
A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n = 7, p = 0.45, x = 6 What is P(6)?
P(6) = .032 use formula: P(x) = nCx p^x (1 − p)^n−x P(6) = [ 7! / 6! (7 - 6)! ] (.45)^6 (.55)^1 = .032
Let the sample space be S= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Suppose the outcomes are equally likely. Compute the probability of the event E= {5, 8}.
P(E) = .2 Out of the values, 5 and 8 occur once each. Therefore, there is a 2/10 chance you will get one of those values
Let the sample space be S= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Suppose the outcomes are equally likely. Compute the probability of the event E="an even number less than 10."
P(E) = .4 Values less than 10 and even are: 2, 4, 6, 8 which is 4/10 given in the set and therefore .4
Determine whether the scatter diagram indicates that a linear relation may exist between the two variables. If the relation is linear, determine whether it indicates a positive or negative association between the variables. Use this information to answer the following. Graph Description: Dots are scattered greatly and possibly show a positive correlation but it does not follow a line. a. Do the two variables have a linear relationship? b. If the relationship is linear do the variables have a positive or negative association?
a. The data points do not have a linear relationship because they do not lie mainly in a straight line. b. The relationship is not linear
Determine whether the scatter diagram indicates that a linear relation may exist between the two variables. If the relation is linear, determine whether it indicates a positive or negative association between the variables. Use this information to answer the following. Graph Description: Dots generally increase as you move right, a line could be drawn in the path a. Do the two variables have a linear relationship? b. If the relationship is linear do the variables have a positive or negative association?
a. The data points have a linear relationship because they lie mainly in a straight line b. The two variables have a positive association.
The word or in probability implies that we use the ??? Rule.
addition
Two events E and F are ??? if the occurrence of event E in a probability experiment does not affect the probability of event F.
independent
